Strong gravitational lensing by irregular mass distributions , such as galaxy clusters , is generally not well quantified by cross sections of analytic mass models .
Computationally expensive ray-tracing methods have so far been necessary for accurate cross-section calculations .
We describe a fast , semi-analytic method here which is based on surface integrals over high-magnification regions in the lens plane and demonstrate that it yields reliable cross sections even for complex , asymmetric mass distributions .
The method is faster than ray-tracing simulations by factors of \sim 30 and thus suitable for large cosmological simulations , saving large amounts of computing time .
We apply this method to a sample of galaxy cluster-sized dark matter haloes with simulated merger trees and show that cluster mergers approximately double the strong-lensing optical depth for lens redshifts z _ { \mathrm { l } } \gtrsim 0.5 and sources near z _ { \mathrm { s } } = 2 .
We believe that this result hints at one possibility for understanding the recently detected high arcs abundance in clusters at moderate and high redshifts , and is thus worth further studies .